Crossing over occurs between equivalent portions of two nonsister chromatids.
Each chromatid contains a single molecule of DNA. So the problem of crossing over is really a problem of swapping portions of adjacent DNA molecules.
It must be done with great precision so that neither chromatid gains or loses any genes. In fact, crossing over has to be sufficiently precise that not a single nucleotide is lost or added at the crossover point if it occurs within a gene. Otherwise a frameshift would result and the resulting gene would produce a defective product or, more likely, no product at all.